Optimal. Leaf size=18 \[ -\frac {2}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {629} \begin {gather*} -\frac {2}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int \frac {b+2 c x}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2}{3 \left (a+b x+c x^2\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 17, normalized size = 0.94 \begin {gather*} -\frac {2}{3 (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} -\frac {2}{3 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 51, normalized size = 2.83 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x + a}}{3 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.99, size = 14, normalized size = 0.78 \begin {gather*} -\frac {2}{3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.70, size = 58, normalized size = 3.22 \begin {gather*} - \frac {2}{3 a \sqrt {a + b x + c x^{2}} + 3 b x \sqrt {a + b x + c x^{2}} + 3 c x^{2} \sqrt {a + b x + c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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